Broadband multimode horn antenna

ABSTRACT

The apparatus of the present invention provides a horn antenna capable of generating a pattern with rotational symmetry and polarization purity over a broadband of frequencies with comparatively low side lobes. In achieving this operation, a first mode which propagates as a &#39;&#39;&#39;&#39;slow wave&#39;&#39;&#39;&#39; is launched through a first portion of the horn and is converted to a second mode which propagates as a &#39;&#39;&#39;&#39;fast wave&#39;&#39;&#39;&#39; through the remaining portion. At the aperture of the horn, the resulting wave combines with a &#39;&#39;&#39;&#39;medium wave&#39;&#39;&#39;&#39; which normally propagates along the entire length of the horn, to achieve the desired aperture distribution. Since the slow wave and the fast wave compensate each other for frequency changes, the overall desired aperture distribution can be sustained over a broad range of frequencies.

United States Patent 3,205,498 9/ 1965 Child lnventor James S. Ajioka Fullerton, Calif.

Appl. No. 771,178

Filed Oct. 28, 1968 Patented Apr. 6, 1971 Assignee Hughes Aircraft Company Culver City, Calif.

BROADBAND MULTIMODE HORN ANTENNA 1 1 Claims, 14 Drawing Figs.

US. Cl 343/783, 343/786 Int. Cl H0lq 13/00 Field of Search 343/772, 776, 777, 778, 779, 786, 783

References Cited UNITED STATES PATENTS 3,413,642 11/1968 Cook Primary Examiner-Eli Lieberman Attorneys-James K l-lakell and Robert H. l-limes ABSTRACT: The apparatus of the present invention provides a horn antenna capable of generating a pattern with rotational symmetry and polarization purity over a broadband of frequencies with comparatively low side lobes. in achieving this operation, a first mode which propagates as a slow wave is launched through a first portion of the horn and is converted to a second mode which" propagates as a fast wave through the remaining portion. At the aperture of the horn, the resulting wave combines with a medium wave which normally propagates along the entire length of the horn, to achieve the desired aperture distribution. Since the slow wave and the fast wave compensate each other for frequency changes, the overall desired aperture distribution can be sustained over a broad range of frequencies.

Patented April 6, 1971 6 Sheets-Sheet 2 Patented April 6, 197-1 6 Sheets-Sheet 4 A iana [dz/Inn: an [K a6 Patented April 6, 1971 6 Sheets-Sheet 5 Patented April 6, 1971 6 Sheets-Sheet 6 1 BROADBAND MULTIMODE HORN ANTENNA BACKGROUND OF THE INVENTION Contemporary single conical or pyramidal horn's which combine higher order modes with a dominant mode can give a variety of radiation patterns. Good control of the radiation pattern can be achieved by controlling the relative amplitudes and phases of the various modes at the horn aperture. High efticiency feed horns for cassegrain antennas utilize this technique. The principal disadvantage to contemporary multimode horns is that the bandwidth is extremely narrow. The reason for this narrow bandwidth is that the higher order modes are generated prior to or in the region of the throat of the horn and since various modes have different velocities of propagation through this region, the relative phases at the horn aperture will vary rapidly with frequency. The longer the horn, the greater is this frequency sensitivity.

One present method of generating higher order modes is the use of a step discontinuity followed by a phasing section to adjust the relative phases of the modes. The relative amplitudes are determined by the height of the step discontinuity and the length of the phasing section is adjusted to give the proper relative phase at the horn aperture. Since the higher order modes have phase velocities that vary much more rapidly with frequency change than the dominant mode, particularly in the phasing section and in the throat region of the horn, the radiation pattern will also change rapidly with changes in frequency. That is, such horns are narrow band devices. Other methods of generating higher order modes are series stub mode generators, and pins or corrugations for mode generation. These methods are narrow band for the same reason as above.

SUMMARY OF THE INVENTION In accordance with the present invention, a plurality of spaced longitudinal conductive rods are disposed in the throat section of a conical horn which as a cross-sectional shape that can be circular, rectangular, triangular, etc. These conductive rods may be parasitically excited or driven externally from the horn to allow remote control of the relative phase and amplitude of the modes to achieve a desired field distribution pattern across the aperture of the horn. In instances where the longitudinal conductive rods are driven externally, hybrid junctions may be used to control the amplitude and phase. Also, for efficient launching of the higher order modes from the TEM lines, the impedance match in the transition from TEM to horn modes must be good. The impedance match can be enhanced by adjustment of the parameters of conductor position in the horn, conductor size and shape, any or all of which can vary along the lines.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 shows a conical multimode horn of circular cross section having parasitically energized longitudinal rods;

FIGS. 2A and 2B illustrate field distribution in the horn of FIG. 1;

FIG. 3 shows a circular multimode horn with externally controlled mode amplitudes and phases;

FIG. 4 shows a circular multimode horn illustrating at TM,, mode field configuration launched from TEM line;

FIGS. 5A and 5B show a circular multimode horn adapted for multimode generation with multiconductor TEM lines excited by a hybrid network;

FIG. 6 shows a conical multimode horn having longitudinal rods with impedance matching;

FIGS. 7A and 7B show a rectangular horn with four longitudinal rods for mode launching;

FIG. 8 shows a cross section of the throat portion of a rectangular horn having a plurality of longitudinal rods for mode launching; I

FIG. 9 shows an E-plane aperture distribution from a TEM launched multimode conical horn;

FIG. 10 shows an H-plane aperture distribution from a TEM launched multimode conical horn; and

FIG. II shows a diagonal-plane aperture distribution from a TEM launched multimode conical horn.

DESCRIPTION Referring now to FIG. 1 of the drawings, there is shown an embodiment of the present invention including a conical horn 10 having a throat section 12 which is terminated with a flange l4. Longitudinal conductive rods 15, 16 are attached diametrically opposite each other to the inner surface and at the right extremity of the throat section 12, as viewed in the drawing, by means of right angle extensions of length h Thus, the rods 15, 16 commence substantially at the termination of the throat section 12 a distance, 11, from the surface and extend for a distance, L,, towards the aperture of the conical horn 10. Although not critical, the rods may extend along lines emanating. from the vertex of conical horn 10 and intersecting points a distance, h, from opposite sides of the throat section 12 at the junction with the conical horn 10.

In the operation of the multimode circular conical horn of FIG. 1, a dominant TE,, mode is initially launched through the throat section 12. The incident TE,, mode excites the TEM mode on the two conductor lines formed by longitudinal rods 15, 16 at the commencement thereof. The magnitude of the TEM mode is controlled by the antenna height, h, i.e., the support portions of the rods 15, 16. Since only a portion of the TEM mode is converted to the TEM made. bqthth lhta t M dessxt t We region L, of the conical horn 10. The configuration of the TEM mode in the region L, in relation to the rods 15, 16 is illustrated in FIG. 2A. At the right extremity of the rods 15, 16, as viewed in the drawing, a TM,, mode is launched by the TEM mode. Thus, the TE,, and TM,, modes exist in the region L of the conical horn 10. Since the longitudinal rods 15, I6 terminated at the end of region L,, the TEM mode cannot exist in the region L The TE,, and TM,, modes combine at the aperture of the horn 10 as illustrated in FIG. 28 to generate a field intensity pattern having very low side lobes. E, H and diagonal plane patterns generated by a TEM launched TM,, mode are illustrated in FIGS. 9, l0 and 11. The side lobes in all these patterns are of the order of 30 db. down from the peak, the patterns are essentially identical in shape showing rotational symmetry and finally, the cross polarization as measured in the diagonal plane where it is greatest is seen to be entirely negligible which illustrates the desireable property of polarization purity.

The relative phase of the TE,, and TM,, modes at the aperture of the born 10 are retained over a broad frequency range. The reason for this is because the phase of the TM,, mode at the horn aperture depends on the relatively slow phase velocity of the TEM mode over the distance L, and the relatively fast phase velocity of the TM,, mode over the distance L, and the phase of the TE,, mode at the horn aperture is a function of the medium velocity of the TE,, mode only over the distance L,+L that is, the composite average phase velocity of a slow wave and a fast wave can be made essentially equal to the average phase velocity of a medium velocity wave over a broad frequency'range thus assuring the relative phase between the TE,, and TM,, waves at the horn aperture to be essentially constant over a broad frequency range (eg., over the entire frequency band of normal operation of a waveguide). Stated mathematically:

L2 7 L2 BTEM Li+f sm au=f fi dl (1) flT =wavelength of TEM mode wavelength of TM mode X =wavelength of Thi mode and phase velocity, v,,' of the modes are 27l' frequency In the equation (1) above, the terms to the left of the equal sign represent the phase of the TM wave at the aperture of coaxial inputs 44, 46 are fed with signals of opposite phase and of a frequency sufficiently high to enable the conical section of the horn to support the TM mode of propagation. Under these circumstances, a TEM mode propagates along the longitudinal rods 47, 48 and launches at TM mode at the terminations thereof. The electric field configurations of the TM mode is illustrated by the lines 55.

the horn 10 while the term to the right of the equal sign represents the phase of the TE, wave at the aperture.

Changes in the frequency of the TEM and TM r nodes are self-compensating, i.e., as one increases the other decreases relative to the TE mode, whereby equation (1) can remain valid for a broad range of frequencies. In general, the slow TEM wave over the distance L, plus the fast TM wave over the distance 1. can equal the medium speed TE wave over the distance L,+84 L and this equality is essentially maintained over a broad frequency range.

Another embodiment of the multimode horn of the present invention adapted to allow external control of the amplitude and phase of the modes is shown in FIG. 3. Referring to FIG. 3, there is shown, in partial section, a conical horn 22 including a throat section 23 with an arm 24. Longitudinal conductive rods 25, 26 extend from the center conductors 27, 28,

respectively, of coaxial lines 29, 30 through throat section 23 after which they fan out along the inner surface of conical horn 22. The end of throat section 23 of horn 22 is terminated by a conductive disc 30 which serves as a termination for the outer conductors of the coaxial lines 29, 30. Coaxial lines 29,

30 are connected to the opposite polarity outputs of a hybrid junction 32 which, in turn, is connected through a variable phase shifter 33 to one output of a variable power divider 34. A remaining output from the variable power divider 34 is coupled to the arm 24 of horn 22 in manner to launch a TE wave therein. An input 36 to the variable power divider 34 serves as an input to the multimode horn 22. Lastly, a remaining output 37 from the hybrid junction 32 is terminated by means of an impedance 38, or may be used as an odd mode generator for an error (difference) channel for monopulse operation.

In operation, a microwave signal is applied to the input line 36 which divides it between the arm 24 and the hybrid junction 32. The hybrid junction 32 applies opposite polarity signals through the coaxial lines 29, 30 to the longitudinal rods in which is directed through the horn 22 in the same manner as the distance L,+I. in the horn 10 of FIG. 1. The relative amplitude and phase of the TE and TM waves at the apertui'e may be controlled to achieve a desired pattern by the variable power divider 34 and the variable phase shifter 33.

Referring 6 FIG. 4, there is shown a multimode horn in accordance with the invention illustrating the electric field configuration for a TM mode. In particular, the apparatus of FIG. 4 illustrates a conical horn 40 in partial section with a throat section 41 terminated by a conductive disc 42. Conductive disc 42 supports coaxial inputs 44, 46, the center conductors of which connect to longitudinal rods 47, 48. Longitudinal rods extend longitudinally through the throat section 41 and then fan but slightly in the conical section of the horn 10.

disc 51 is connected to the throat section 4. In operation, the

Referring to FIGS. 5A and 5B there is shown a multimode horn in accordance with the invention with more than two TEM mode launching rods. In particular, FIGS. 5A and 5B show a conical horn 60 with longitudinal TEM mode launching rods 61, 62, 63, 64. The rods 61-64 are tapered and spacing flared such that the characteristic impedance for the TEM mode is constant (i.e. the ratio of rod diameter and spacing to horn diameter are constant to give geometrical similarity at any cross section). In particular, FIGS. 5A and 5B show a conical horn 60 with longitudinal TEM mode launching rods 61, 62, 63, 64 shown in cross section in FIG. 5B. The longitudinal rods 61, 62, 63, 64 are driven by means of a hybrid feed network 65 through coaxial lines 66, 67, 68, 69. In operation, the hybrid feed network 65 excites the multiconductor TEM lines 61-64 with signals of the proper amplitudes and phases which, in turn, launch the higher order modes in the horn 60. The relative phase of the signals applied to the multiconductor TEM lines 61-64 proceeding in a clockwise direction as viewed in FIG. 5B is 2'n'm/n where m32 i0, 1, 2, 3... and n is the number of lines. Thus, for m=l in the present case, the relative phase between adjacent lines is 90. This phase difference is employed for exciting a TEM mode which, in turn, launches dual orthogonal (right and left circularly polarized) modes in the horn 60.

For efficient launching of the higher order modes from the TEM lines or longitudinal rods, the impedance match in the transition from TEM to horn modes must be good. Referring to FIG. 6, there is shown a conical horn including longitudinal rods 71, 72. Conductive ferrules 73, 74 are disposed near the end of the 'rods 71, 72, respectively, and ferrules 75, 76 at the extremities thereof. The ferrules 73, 74 are of the order of twice the length of the ferrules 75, 76 and are spaced of the order of one length away therefrom. In addition, the diameter of ferrules 73, 74, 75, 76 is of the order of two to three times the diameter of the rods 71, 72. The same configuration could, of course, be achieved by undercutting a heavier rod. The principal consideration to achieve impedance matching is to generate reflections to cancel out reflections developed at the transition between the rods 71, 72 and free space within the horn 70.

Longitudinal rods can, of course, be used with horns of other shapes such as horns with rectangular or triangular cross section. Referring to FIGS. 7A and 73 there is shown a rectangular horn 80 with longitudinal rods 81, 82, 83, 84. Connections to the rods 81-84 are made through coaxial connectors 85. A more complex pattern of longitudinal rods in a rectangular horn 86 is shown in cross section in FIG. 8, wherein there are two horizontal rows 87, 88 and two vertical rows 89, 90, as viewed in the drawing, of six rods each. The height of the horn 86 is b, the width 0, the vertical distance from the lower side, as viewed in the drawing, y, and the horizontal distance from the left side, as viewed in the drawings, x. In general, the rows 87-90 of rods can be driven in a manner to achieve any desired pattern as might be determined by a finite Fourier series. I'Iigher are modes and launched, however, by driving the discrete rods in accordance with corresponding points of the desired field intensity pattern within the rectangular horn 86. In this latter case, excitation of rods in rows 87, 88 is in accordance with discrete sampling of cos(mrx/a) where x denotes position of rod in x direction. Similarly, excitation of rods in rows 89, 90 is in accordance with discrete sampling of cos(m1ry/b) where y denotes position or rod in y direction. In the foregoing situation, m and n are integers which denote the particular mode to be launched. The rods 81-84 in the horn 80 of FIGS. 7A and 7B are excited in accordance with the manner described in connection with FIG. 8.

lclaim:

1. A broadband multimode horn antenna comprising a conductive horn having a conical configuration, means for launching and propagating a first electromagnetic wave in the TE mode of predetermined velocity through first and second successive portions of the length of said conductive horn, means for launching and propagating a second electromagnetic wave in the TEM mode having a velocity slower than said predetermined velocity through said first portion of the length of said conductive horn, and means for converting said second wave into a third electromagnetic wave in the TM mode having a velocity faster than said predetermined velocity through said second portion of the length of said conductive horn whereby said first and third waves combine to provide a predetermined field intensity pattern at the aperture of said horn over a broad range of frequencies.

2. A broadband multimode horn antenna comprising a conductive horn having an aperture at one extremity thereof; a plurality of longitudinal conductive spaced rods disposed along a first portion of the length of said horn spaced from said aperture; means coupled to said horn for launching a wave of predetermined velocity therethrough; and means coupled to said spaced rods for launching a TEM wave therealong of a velocity less than said predetermined velocity whereby said TEM wave converts to a wave of a velocity faster than said predetermined velocity at the termination of said first portion and propagates through a second portion of the length of said horn extending from said first portion to said aperture to combine with said wave of predetermined velocity to provide a predetermined field intensity pattern.

3. The broadband multimode horn as defined in claim 2 wherein said conductive horn has a conical configuration with a constant diameter throat section, and said plurality of iongitudinal conductive spaced rods constitutes first and second rods disposed opposite each other along said first portion of the length of said horn.

4. The broadband multimode horn antenna as defined in claim 2 wherein the ratio of the crosssectional dimensions of the diameter of said rods and the relative spacing thereof to the dimensions of said horn at said cross section are constant thereby to provide a constant impedance to said TEM wave.

5. The broadband multimode horn as defined in claim 2 wherein said conductive horn has a conical configuration with a constant diameter throat section, and said plurality of lon gitudinal conductive spaced rods constitutes first, second, third and fourth rods disposed at quadrature points along said first portion of the length of said horn.

6. The broadband multimode horn as defined in claim 2 wherein the respective end portions of said plurality of spaced longitudinal conductive rods nearest said aperture each include no less than one step discontinuity therealong thereby to provide impedance matching between said rods and said second portion of the length of said horn.

7. A broadband multimode horn antenna comprising a conical horn having a cylindrical throat section at the narrow extremity thereof and an aperture at the remaining extremity; first and second longitudinal rods disposed along a portion of the length of said conical horn adjacent said throat section on opposite sides of the centerline thereof, said longitudinal rods being supported by right-angle extensions thereof at the extremities nearest said throat section to the inner surface of said conical horn; and means for launching a TE wave through said throat section thereby to generate a multimode field intensity pattern at said aperture.

8. A broadband multimode horn antenna system comprising a conical horn having a cylindrical throat section at the narrow extremity thereof and an aperture at the remaining extremity; first and second longitudinal rods disposed through said throat section and along a portion of the length of said conical horn adjacent thereto on opposite sides of the centerline thereof; means responsive to a signal to be radiated from said horn for dividing the power and controlling the relative phase thereof along first and second paths, said first path being coupled to said throat section for exciting a TE wave therethrough; and means including a hybrid junction having an input connected to second path and outputs connected to said first and second rods for launching said signal as a TEM wave therealong thereby to generate a controllable field intensity pattern representative of said signal at said aperture.

9. A broadband multimode horn antenna system comprising a conical horn having a cylindrical throat section at the narrow extremity thereof and an aperture at the remaining extremity; a plurality, n, of longitudinal conductive rods disposed through said throat section and along a portion of the length of said conical horn adjacent thereto at equal intervals about the centerline thereof; and a hybrid feed network responsive to a signal to be radiated from said aperture and connected to said n longitudinal conductive rods for exciting said rods with said signal with a phase difference between successive rods equal to cos(21rm/n) where m is an integer.

10. A broadband multimode horn antenna comprising a rectangular horn having a rectangular throat section at the narrow extremity thereof and a rectangular aperture at the remaining extremity; a plurality of conductive rods disposed lon itudinally throu h said throat section and along a portion of t e length of san rectangular horn ad acent there 0; and

means connected to said rods for generating a predetermined multimode field intensity pattern at said rectangular aperture.

11. The broadband multimode horn antenna as defined in claim 10 wherein said rectangular throat section has a width a and a height b and wherein x is a variable denoting the respec tive positions of a portion of said rods along said width from a first reference side of said throat section and y is a variable denoting the respective positions of the remainder of said rods along said height from a second reference side of said throat section orthogonal to said first reference side and wherein said last-named means constitutes discrete sampling of cos(nrrm/a along said width for said portion of said rods and constitutes discrete sampling of cos(m1ry/b) along said heights for said remainder of said rods wherein m and n are integers that determine the modes of propagation.

mg UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION Patent No. 3,573,838 Dated April 6, 1971 Inventor-(s) Iames S Ajioka It is certified that error appears in the above-identified patent and that said Letters Patent are hereby corrected as shown below:

r- Column 3, line 6 ."BTEM wavelength of TEM mode" should be ATEM wavelength of TEM mode-- Column 6, line 53, "cos(mr m/a" should be cos(mrx/a)-- Signed and sealed this 8th day of May 1973.

(SEAL) Attest:

EDWARD M.FLETCHER,JR. ROBERT GOTTSCHALK Attesting Officer Commissioner of Patent: 

1. A broadband multimode horn antenna comprising a conductive horn having a conical configuration, means for launching and propagating a first electromagnetic wave in the TE11 mode of predetermined velocity through first and second successive portions of the length of said conductive horn, means for launching and propagating a second electromagnetic wave in the TEM mode having a velocity slower than said predetermined velocity through said first portion of the length of said conductive horn, and means for converting said second wave into a third electromagnetic wave in the TM11 mode having a velocity faster than said predetermined velocity through said second portion of the length of said conductive horn whereby said first and third waves combine to provide a predetermined field intensity pattern at the aperture of said horn over a broad range of frequencies.
 2. A broadband multimode horn antenna comprising a conductive horn having an aperture at one extremity thereof; a plurality of longitudinal conductive spaced rods disposed along a first portion of the length of said horn spaced from said aperture; means coupled to said horn for launching a wave of predetermined velocity therethrough; and means coupled to said spaced rods for launching a TEM wave therealong of a velocity less than said predetermined velocity whereby said TEM wave converts to a wave of a velocity faster than said predetermined velocity at the termination of said first portion and propagates through a second portion of the length of said horn extending from said first portion to said aperture to combine with said wave of predetermined velocity to provide a predetermined field intensity pattern.
 3. The broadband multimode horn as defined in claim 2 wherein said conductive horn has a conical configuration with a constant diameter throat section, and said plurality of longitudinal conductive spaced rods constitutes first and second rods disposed opposite each other along said first portion of the length of said horn.
 4. The broadband multimode horn antenna as defined in claim 2 wherein the ratio of the cross-sectional dimensions of the diameter of said rods and the relative spacing thereof to the dimensions of said horn at said cross section are constant thereby to provide a constant impedance to said TEM wave.
 5. The broadband multimode horn as defined in claim 2 wherein said conductive horn has a conical configuration with a constant diameter throat section, and said plurality of longitudinal conductive spaced rods constitutes first, second, third and fourth rods disposed at quadrature points along said first portion of the length of said horn.
 6. The broadband multimode horn as defined in claim 2 wherein the respective end portions of said plurality of spaced longitudinal conductive rods nearest said aperture each includE no less than one step discontinuity therealong thereby to provide impedance matching between said rods and said second portion of the length of said horn.
 7. A broadband multimode horn antenna comprising a conical horn having a cylindrical throat section at the narrow extremity thereof and an aperture at the remaining extremity; first and second longitudinal rods disposed along a portion of the length of said conical horn adjacent said throat section on opposite sides of the centerline thereof, said longitudinal rods being supported by right-angle extensions thereof at the extremities nearest said throat section to the inner surface of said conical horn; and means for launching a TE11 wave through said throat section thereby to generate a multimode field intensity pattern at said aperture.
 8. A broadband multimode horn antenna system comprising a conical horn having a cylindrical throat section at the narrow extremity thereof and an aperture at the remaining extremity; first and second longitudinal rods disposed through said throat section and along a portion of the length of said conical horn adjacent thereto on opposite sides of the centerline thereof; means responsive to a signal to be radiated from said horn for dividing the power and controlling the relative phase thereof along first and second paths, said first path being coupled to said throat section for exciting a TE11 wave therethrough; and means including a hybrid junction having an input connected to second path and outputs connected to said first and second rods for launching said signal as a TEM wave therealong thereby to generate a controllable field intensity pattern representative of said signal at said aperture.
 9. A broadband multimode horn antenna system comprising a conical horn having a cylindrical throat section at the narrow extremity thereof and an aperture at the remaining extremity; a plurality, n, of longitudinal conductive rods disposed through said throat section and along a portion of the length of said conical horn adjacent thereto at equal intervals about the centerline thereof; and a hybrid feed network responsive to a signal to be radiated from said aperture and connected to said n longitudinal conductive rods for exciting said rods with said signal with a phase difference between successive rods equal to cos(2 pi m/n) where m is an integer.
 10. A broadband multimode horn antenna comprising a rectangular horn having a rectangular throat section at the narrow extremity thereof and a rectangular aperture at the remaining extremity; a plurality of conductive rods disposed longitudinally through said throat section and along a portion of the length of said rectangular horn adjacent thereto; and means connected to said rods for generating a predetermined multimode field intensity pattern at said rectangular aperture.
 11. The broadband multimode horn antenna as defined in claim 10 wherein said rectangular throat section has a width a and a height b and wherein x is a variable denoting the respective positions of a portion of said rods along said width from a first reference side of said throat section and y is a variable denoting the respective positions of the remainder of said rods along said height from a second reference side of said throat section orthogonal to said first reference side and wherein said last-named means constitutes discrete sampling of cos(n pi m/a) along said width for said portion of said rods and constitutes discrete sampling of cos(m pi y/b) along said heights for said remainder of said rods wherein m and n are integers that determine the modes of propagation. 